Let C(X,E) denote the set of all continuous functions
from a topological space X into a topological space E.
R. Engelking and S. Mrowka [2] proved that for any E-completely
regular space X [Definition 1.1], there exists a unique E-compactification
[formula omitted] [Definitions 2.1 and 3.1] with the property
that every function f in C(X,E) has an extension f in [formula omitted].
It is proved that if E is a (*)-topological division
ring [Definition 5-5] and X is an E-completely regular space,
then [formula omitted] is the same as the space of all E-homomorphisms
[Definition 5.3] from C(X,E) into E. Also, we establish that
if E is an H-topological ring [Definition 6.1] and X, Y are
E-compact spaces [Definition 2.1], then X and Y are homeomorphic
if, and only if, the rings C(X,E) and C(Y,E) are E-isomorphic
[Definition 5.3]. Moreover, if t is an E-isomorphism from
C(X,E) onto C(Y,E) then [formula omitted] is the unique homeomorphisms
from Y onto X with the property that [formula omitted] for all
f in C(X,E), where π is the identity mapping on X and t
is a certain mapping induced by t. In particular, the development
of the theory of C(X,E) gives a unified treatment for the cases
when E is the space of all real numbers or the space of all
integers.
Finally, for a topological ring E, the bounded subring
C*(X,E) of C(X,E) is studied. A function f in C(X,E) belongs
to C*(X,E) if for any O-neighborhood U in E, there exists a 0-neighborhood V in E such that f[X]•V c U and V•f[X] c U.
The analogous results for C*(X,E) follow closely the theory of
C(X,E); namely, for any E*-completely regular space X
[Definition 9.5], there exists an E*-compactification [formula omitted] of
X such that every function f in C (X,E) has an extension f
in [formula omitted] when E is the space of all nationals, real numbers,
complex numbers, or the real quaternions, [formula omitted] is just the space
of all E-homomorphisms from C*(X,E) into E. This is also valid
for a topological ring E which satisfies certain conditions. Also,
two E*-compact spaces [Definition 10.1] X and Y are homeomorphic
if, and only if, the rings C*(X,E) and C*(Y,E) are E-isomorphic, where E is any H*-topological ring [Definition 12.8]. / Science, Faculty of / Mathematics, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/41205 |
| Date | January 1969 |
| Creators | Chew, Kim-Peu |
| Publisher | University of British Columbia |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
Page generated in 0.0022 seconds